Estimation of the local intensity of a cyclic Poisson process by means of nearest neighbor distances

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

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Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions

Advances in Applied Probability ◽

10.2307/1427088 ◽

1985 ◽

Vol 17 (4) ◽

pp. 794-809 ◽

Cited By ~ 5

Author(s):

Charles M. Newman ◽

Yosef Rinott

Keyword(s):

Asymptotic Behavior ◽

Point Process ◽

Poisson Point Process ◽

Point Processes ◽

Nearest Neighbor ◽

Correlation Coefficients ◽

Nearest Neighbors ◽

Distance Functions ◽

High Dimensional ◽

Voronoi Region

Consider a Poisson point process of density 1 in Rd, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e0, · ··, en, in Rd and a variety of distances, we obtain the asymptotic behavior of Ndn, the number of points which have e0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-lp distances related to correlation coefficients.

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Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions

Advances in Applied Probability ◽

10.1017/s000186780001541x ◽

1985 ◽

Vol 17 (04) ◽

pp. 794-809 ◽

Cited By ~ 4

Author(s):

Charles M. Newman ◽

Yosef Rinott

Keyword(s):

Asymptotic Behavior ◽

Point Process ◽

Poisson Point Process ◽

Point Processes ◽

Nearest Neighbor ◽

Correlation Coefficients ◽

Nearest Neighbors ◽

Distance Functions ◽

High Dimensional ◽

Voronoi Region

Consider a Poisson point process of density 1 in R d, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e 0, · ··, e n, in Rd and a variety of distances, we obtain the asymptotic behavior of Nd n , the number of points which have e 0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-l p distances related to correlation coefficients.

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Properties of the bivariate delayed Poisson process

Journal of Applied Probability ◽

10.1017/s002190020004794x ◽

1975 ◽

Vol 12 (02) ◽

pp. 257-268 ◽

Cited By ~ 1

Author(s):

A. J. Lawrance ◽

P. A. W. Lewis

Keyword(s):

Poisson Process ◽

Point Process ◽

Joint Distribution ◽

Poisson Point Process ◽

Initial Conditions ◽

Arbitrary Time ◽

Delay Distribution ◽

Counting Distribution

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

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Focusing of the scan statistic and geometric clique number

Advances in Applied Probability ◽

10.1239/aap/1037990951 ◽

2002 ◽

Vol 34 (4) ◽

pp. 739-753 ◽

Cited By ~ 5

Author(s):

Mathew D. Penrose

Keyword(s):

Poisson Process ◽

Point Process ◽

Poisson Point Process ◽

Dimensional Space ◽

Clique Number ◽

Geometric Graph ◽

Finite Range ◽

Scan Statistic ◽

Constant Intensity ◽

Binomial Point Process

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

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L2-CONVERGENCE OF A NEAREST NEIGHBOR ESTIMATOR OF THE INTENSITY FUNCTION OF A CYCLIC POISSON PROCES

Journal of Mathematics and Its Applications ◽

10.29244/jmap.13.2.49-62 ◽

2014 ◽

Vol 13 (2) ◽

pp. 49

Author(s):

I W. MANGKU

Keyword(s):

Poisson Process ◽

Key Words ◽

Nonparametric Estimation ◽

Nearest Neighbor ◽

Intensity Function ◽

Key Words And Phrases ◽

Subject Classifications ◽

Single Realization ◽

Func Tion

<p>Abstract. We consider the problem of estimating the intensity func- tion of a cyclic Poisson process. We suppose that only a single realization of the cyclic Poisson process is observed within a bounded 'window', and our aim is to estimate consistently the intensity function at a given point. A nearest neighbor estimator of the intensity function is proposed, and we show that our estimator is L2-consistent, as the window expands.<br />AMS 2010 subject classifications: 62E20, 62G05, 62G20.<br />Key words and phrases: cyclic Poisson process, cyclic intensity function, nonparametric estimation, nearest neighbor estimator, period, consis- tency, L2-convergence.</p>

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Properties of the bivariate delayed Poisson process

Journal of Applied Probability ◽

10.2307/3212439 ◽

1975 ◽

Vol 12 (2) ◽

pp. 257-268 ◽

Cited By ~ 2

Author(s):

A. J. Lawrance ◽

P. A. W. Lewis

Keyword(s):

Poisson Process ◽

Point Process ◽

Joint Distribution ◽

Poisson Point Process ◽

Initial Conditions ◽

Arbitrary Time ◽

Delay Distribution ◽

Counting Distribution

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

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Focusing of the scan statistic and geometric clique number

Advances in Applied Probability ◽

10.1017/s0001867800011897 ◽

2002 ◽

Vol 34 (04) ◽

pp. 739-753 ◽

Cited By ~ 1

Author(s):

Mathew D. Penrose

Keyword(s):

Poisson Process ◽

Point Process ◽

Poisson Point Process ◽

Dimensional Space ◽

Clique Number ◽

Geometric Graph ◽

Finite Range ◽

Scan Statistic ◽

Constant Intensity ◽

Binomial Point Process

Given sets C and R in d-dimensional space, take a constant intensity Poisson point process on R; the associated scan statistic S is the maximum number of Poisson points in any translate of C. As R becomes large with C fixed, bounded and open but otherwise arbitrary, the distribution of S becomes concentrated on at most two adjacent integers. A similar result holds when the underlying Poisson process is replaced by a binomial point process, and these results can be extended to a large class of nonuniform distributions. Also, similar results hold for other finite-range scanning schemes such as the clique number of a geometric graph.

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Application of Lattice Imaging Techniques to Amorphous Films

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100113664 ◽

1975 ◽

Vol 33 ◽

pp. 8-9

Author(s):

S. R. Herd ◽

P. Chaudhari

Keyword(s):

Nearest Neighbor ◽

Random Network ◽

Imaging Techniques ◽

Network Models ◽

Accurate Method ◽

Direct Transmission ◽

Lattice Imaging ◽

Transmission Electron ◽

Lattice Planes ◽

Nearest Neighbor Distances

Electron diffraction and direct transmission have been used extensively to study the local atomic arrangement in amorphous solids and in particular Ge. Nearest neighbor distances had been calculated from E.D. profiles and the results have been interpreted in terms of the microcrystalline or the random network models. Direct transmission electron microscopy appears the most direct and accurate method to resolve this issue since the spacial resolution of the better instruments are of the order of 3Å. In particular the tilted beam interference method is used regularly to show fringes corresponding to 1.5 to 3Å lattice planes in crystals as resolution tests.

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